Encyclopedia of Microtonal Music Theory

monzo / exponent vector / prime-exponent vector

[Joe Monzo]

A set of numbers notated in the form [a b, c d e, ... z > , which is the list of exponents of the prime-factors up to the prime-limit z, and which uniquely identifies an interval as a vector in prime-space. It is a specific cardinality referring to the 1-dimensional case of the more general term multimonzo.

A comma punctuation-mark is used after the exponent of 3, and then subsequently after every third exponent (i.e., after the exponents of prime-factors 11, 19, 31, 43, etc.). This makes it easy to identify whether prime-factor 2 is being recognized or ignored, and aids in recognition of higher primes.

A wildcard character * may also be used for any prime or combination of primes which are considered to form an equivalence-interval. Thus, for example, the monzo for the syntonic-comma (ratio 81:80) in the usual case where prime-factor 2 is ignored, is [ * 4, -1 > . (see Robert Walker, Yahoo tuning group, message 55056 ... which contains some errors.)

The standard monzo is defined as admitting only integer exponents; a rational monzo allows rational exponents as well (... many examples of the latter may be found in this Encyclopaedia's entries describing temperaments such as 1/4-comma meantone).

A monzo may be thought of as a set of "driving directions" thru prime-space along the taxicab metric of a lattice diagram. Geometrically, a monzo may usually be thought of as a Euclidean directed line-segment in tone-space (prime-space) unless it vanishes, in which case it becomes a point.

The term was coined in July 2003 by Gene Ward Smith in honor of Joe Monzo's advocacy of prime-factor vectors as part of the musical notation, in Monzo 1997, JustMusic Prime-Factor Notation.

A special case of the monzo is when it is tempered-out (or "vanishes"), in which case it is represented by, or is, a promo.

For p an odd prime, the intervals of the p-limit Np may be taken as the set of all frequency ratios which are positive rational numbers whose factorization involves only primes less than or equal to p.

If q is such a ratio, it may be written in factored form as

q = 2e2 3e3 ... pep

where e2, e3, ... ep are integer exponents.

We may write this in factored form as a row vector of the exponents, or monzo:

[e2, e3, ..., ep]

The p-limit rational numbers Np form an abelian group, or Z-module, under multiplication, so that it acts on itself as a transformation group of a musical space; this becomes an additive group using vector addition when written additively as a monzo.

Physicists sometimes use bra and ket vectors; if you did that, the monzo for 81/80 would be a ket, [-4 4 -1> and the val for 5-limit 12-et would be a bra, <12 19 28]. Putting them together would give the bra-ket, angle bracket, or inner product: <12 19 28 | -4 4 -1> = 0. See:

In Yahoo tuning-math, message 7435 (Thu Nov 6, 2003 3:48 pm), George Secor proposed using comma punctuation marks after the exponent of prime-factor 3 and then again after every third prime-factor. I accept both this idea and the ket notation proposed by Gene Ward Smith described above. Thus, labeling each exponent with its prime-factor, the standard i propose for the notation of monzos is:

[2 3, 5 7 11, 13 17 19, 23 29 31, 37 41 43, ... etc.>

If there are large gaps in the set of prime-factors, those with a zero-exponent may be omitted, and a comma used to delimit every member of the monzo; the prime-factors must be explicitly stated. (An example of this may be seen the Tuning Encyclopaedia description of the epimoric approximation of the Pythagorean comma.)

Monzos for many of the ratios that might be encountered in 11-limit just intonation, may be found in the "big list of 11-limit intervals" webpage.

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